As we know by the Shuffle Theorem of [1]}, \(\nabla ^ke_n[X]\)  is the character of the diagonal coinvariants, and is the result of substituting \(p_k(Y)=1\)  in \(e_n[XY]\) , in other words taking the trivial component of the representation of the ”dot" action. By [2]}, this is the same as the \(_0\)  part, and so corresponds to tensoring out both \(\mathbf {x}\)  and \(\mathbf {y}\)  \(H_*^T(^{\gamma ,+}_)\)  over \(\Lambda ^+]\otimes ]\cong \mathbf {x},\mathbf {y}]\) , without including higher derived functors.
